Some problems concerning the rigidity of manifolds

In this paper, we systematically review the intrinsic relationships between the volume of manifolds and Sobolev constants, Ricci curvature, or scalar curvature, summarize the research progress on the rigidity of Sobolev constants on manifolds, and outline the classification results for positive solutions of the critical p-Laplace equation defined on non-compact complete manifolds with non-negative Ricci curvature. Based on the current state of the two major research directions mentioned above, we further propose some key open problems.